Burden of proof and Occam’z razor

The person making the claim has the burden of proof. If someone claims they have a cat, it is not practical nor reasonable to demand evidence for that claim, and if someone were to demand evidence, the person would simply say they don't care if they’re believed or not.

The Burden of Proof is only a thing in argumentation. When one's goal is convince another that a claim is true, they have the burden of proof. If the theist claims God exists, they have the burden of proof. If an atheist claims that God does not exist, they have the burden of proof.

First, Occam’s Razor doesn’t apply.  That’s the principle that all other things bring equal, the simplest explanation should be preferred.  So if a phenomenon needs to be explained, a simpler explanation should be considered rather than a more complicated one.  As additional information is obtained, the theory may need to be revised and become more complicated to accommodate the new evidence.  But that has nothing to do. in your example, of who owns the cat. 
Burden of Proof is a legal concept.  The party making the claim has the burden of proof.  So if I claim the cat you’re keeping us mine, I’d have the burden of proving it.